We give a new existence criterion for p-ary sequences which have ideal two-level autocorrelation; and we use it to obtain four general families of such sequences: one for p=2, one for general odd primes p and two special ones for p=3. The binary family turns out to be equivalent to that discovered by Dillon and Dobbertin and published in 2004. The general p-ary family is equivalent to that discovered by Gong and Helleseth, by Dillon and, when p=3, by Helleseth, Kumar, and Martinsen. All of these p-ary results were published in 2001 and 2002. The special ternary families are new and give as special cases the sequences conjectured by Alfred Lin in his 1998 Ph.D. thesis as well as most of those conjectured in 2001 by Ludkovski and Gong. Our sequences may also be used to construct (relative) difference sets, their corresponding block designs and generalized weighing matrices.